A logarithm serves as the inverse function of exponentiation. It represents the power to which a specific number must be raised to yield another given number. Introduced by John Napier in the early 17th century, logarithms aimed to streamline calculations.
Example: log 100 = 2
Taking LHS
log 10^{2}
= 2 log 10
= 2 × 1 (∴ Log 10 = 1)
= 2.
Let’s have a look on some question that will help better in understanding Logarithm.
Basic Difference Between Logarithm and Natural Logarithm:
Q1: Which of the following statement is not correct?
(a) log (1 + 2 + 3) = log 1 + log 2 + log 3
(b) log( 2+3) = log (2×3)
(c) log_{10} 1 = 0
(d) log_{10} 10 = 1
Ans: (b)
(a) log( 1+2+3) = log6 = log (1×2×3) = log1+ log2+log3
(b) log (2 + 3) = log 5 and log (2 x 3) = log 6 = log 2 + log 3
log (2 + 3) ≠ log (2 x 3)
(c) Since, log_{a} 1=0,
so log_{10 }1=0
(d) Since, log_{a} a = 1
so, log_{10} 10 = 1.
Q3: Solve for x: log_{2} (x + 3) + log_{2} (x – 1) = 3
Sol: Combine the logarithms using the logarithm property log(a) + log(b) = log(a * b):
log_{2} ((x + 3)×(x − 1)) = 3
2^{3} =(x+3)×(x−1)
8 = (x + 3) × (x − 1)
8 = x^{2} + 2x − 3
x^{2} + 2x − 11 = 0
(x + 4)(x  2) = 0
x = 4, 2
Q2: If X is an integer then solve(log_{2} X)^{2}– log_{2}x^{4}32 = 0
Sol: Let us consider,(log_{2} X)^{2}– log_{2}x^{4}32 = 0 —— equation 1
let log_{2} x= y
equation 1 = y^{2}– 4y32=0
y^{2}8y+ 4y – 32= 0
y(y8) + 4(y8) =0
(y8) (y+4)= 0
y=8, y= 4
log_{2}X = 8 or, log_{2}X = 4
X = 2^{8} = 256, 0r
Since, X is an integer so X = 256.
314 videos170 docs185 tests

1. What is a logarithm? 
2. Why are logarithms useful in mathematics? 
3. What are the properties of logarithms? 
4. How do logarithms help solve exponential equations? 
5. Can logarithms be negative or zero? 
314 videos170 docs185 tests


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